In this work optical solitary waves are generated through a variational approach using the symbiotyc organism search (SOS) to optimize the parameters of the ansatz functions.
The NLSE is a very well known equation that has been studied in optics. The NLSE equation can be stated in an an arbitrary nonlinear medium, that for the case of two dimensional scenario is
where is the corresponding function that describe the effects of the nonlinear medium. In our physical model,
stands for the complex optical field, r=(x,y) are the transverse spatial coordinates, and z is the longitudinal (propagation) spatial coordinate, and
stands for the transverse Laplacian. Note that the generalized NLSE is written in dimensionless form.
The NLSE can be described as a variational problem with the following Lagrangian density
here is the anharmonic term in the Lagrangian density, where is possible to describe from several nonlinear media.
As we are interested in self-trapped solutions to our optical system, we propose the following ansatz
where
is a real propagation phase constant and U(r) is a function that only depends on the transverse spatial coordinates. Substituting this \textit{ansatz} in the equation the relation can be simplified to
A crucial step in the variational approach forsoliton optimization, that was introduced in the seminal paper by Anderson isto select carefully the function U(x,y). For this work we used a Gaussian beam
where A and b stand for the amplitude and width respectively, and they are parameters that are going to be optimized by SOS.
Next, we search for approximate soliton solutions by solving the corresponding Euler-Lagranage equations; however, for the stationary scenario, we are interested in the saddle points of the functional f
meaning that we search a function g that minimize the following condition
where is the gradient of the
functional with respect to the parameters A and b.
SOS was implemented to find the parameters A and b that make . Since the function only takes positive values it is a minimization problem. The funtion g was used as the fitness function of the algorithm.